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G = C22:D8  order 64 = 26

The semidirect product of C22 and D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D4:2D4, C22:2D8, C23.41D4, (C2xD8):1C2, C2.4(C2xD8), C4:D4:1C2, C22:C8:3C2, C4:C4:1C22, (C2xC8):1C22, C4.19(C2xD4), (C2xC4).22D4, D4:C4:4C2, C2.8C22wrC2, (C2xD4):1C22, (C22xD4):2C2, C2.6(C8:C22), (C2xC4).81C23, C22.77(C2xD4), (C22xC4).42C22, SmallGroup(64,128)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2xC4 — C22:D8
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C22xD4 — C22:D8
Lower central
C1 — C2 — C2xC4 — C22:D8
Upper central
C1 — C22 — C22xC4 — C22:D8
Jennings
C1 — C2 — C2 — C2xC4 — C22:D8

Generators and relations for C22:D8
 G = < a,b,c,d | a2=b2=c8=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 225 in 99 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, D4, C23, C23, C22:C4, C4:C4, C2xC8, D8, C22xC4, C2xD4, C2xD4, C2xD4, C24, C22:C8, D4:C4, C4:D4, C2xD8, C22xD4, C22:D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C22wrC2, C2xD8, C8:C22, C22:D8

Character table of C22:D8

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D8A8B8C8D
 size 1111224444822484444
ρ11111111111111111111    trivial
ρ21111111111-1111-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ4111111-1-1-1-1-1111-11111    linear of order 2
ρ51111-1-1-1-111111-1-11-1-11    linear of order 2
ρ61111-1-1-1-111-111-11-111-1    linear of order 2
ρ71111-1-111-1-1111-1-1-111-1    linear of order 2
ρ81111-1-111-1-1-111-111-1-11    linear of order 2
ρ92-22-200-220002-2000000    orthogonal lifted from D4
ρ102-22-2002-20002-2000000    orthogonal lifted from D4
ρ112222-2-200000-2-2200000    orthogonal lifted from D4
ρ122-22-200002-20-22000000    orthogonal lifted from D4
ρ132-22-20000-220-22000000    orthogonal lifted from D4
ρ1422222200000-2-2-200000    orthogonal lifted from D4
ρ152-2-22-22000000000√2-√2√2-√2    orthogonal lifted from D8
ρ162-2-222-2000000000√2√2-√2-√2    orthogonal lifted from D8
ρ172-2-22-22000000000-√2√2-√2√2    orthogonal lifted from D8
ρ182-2-222-2000000000-√2-√2√2√2    orthogonal lifted from D8
ρ1944-4-4000000000000000    orthogonal lifted from C8:C22

Permutation representations of C22:D8
►On 16 points - transitive group 16T126
Generators in S16
(1 5)(2 9)(3 7)(4 11)(6 13)(8 15)(10 14)(12 16)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)
 
G:=sub<Sym(16)| (1,5)(2,9)(3,7)(4,11)(6,13)(8,15)(10,14)(12,16), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)>;
 
G:=Group( (1,5)(2,9)(3,7)(4,11)(6,13)(8,15)(10,14)(12,16), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14) );
 
G=PermutationGroup([[(1,5),(2,9),(3,7),(4,11),(6,13),(8,15),(10,14),(12,16)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,16),(7,15),(8,14)]])
 
G:=TransitiveGroup(16,126);
 

C22:D8 is a maximal subgroup of
 C23:D8  C4:C4.D4  C24.9D4  C24.103D4  C24.177D4  C24.105D4  C4oD4:D4  (C2xD4):21D4  C42.225D4  C42.227D4  C42.232D4  C42.352C23  C42.356C23  C23:3D8  C24.121D4  C24.125D4  C24.127D4  C4.2+ 1+4  C4.142+ 1+4  C42.269D4  C42.271D4  C42.275D4  C42.406C23  C42.410C23  SD16:D4  SD16:7D4  SD16:1D4  D4xD8  SD16:10D4  D4:4D8  C42.462C23  C42.41C23  C42.53C23  C42.54C23  C42.471C23  C42.474C23  D4:S4
 D4p:D4: D8:9D4  D8:5D4  D12:13D4  D4:D12  D12:16D4  D12:D4  D20:13D4  D4:D20 ...
 (C2xC2p):D8: (C2xC4):D8  C42.221D4  C42.263D4  (C2xC6):8D8  (C2xC10):8D8  (C2xC14):8D8 ...
C22:D8 is a maximal quotient of
 C23:D8  C23.5D8  (C2xC4).5D8  D4:D8  Q8:D8  D4:3D8  Q8:3D8  D4.D8  Q8.D8  D4.7D8  D4:4Q16  C23.35D8  C23.37D8  C2.(C4xD8)  C23.38D8  C23:2D8  (C2xD4):Q8  C24.83D4  C4:C4:7D4  C4:C4:Q8  Q16:7D4  D8.9D4  Q16.8D4  D8.10D4  D8.D4  Q16.10D4  Q16.D4  D8.3D4  D8.12D4
 D4p:D4: D8:7D4  D8:8D4  D8:D4  D12:13D4  D4:D12  D12:16D4  D12:D4  D20:13D4 ...
 (C2xC2p):D8: (C2xC4):D8  (C2xC4):9D8  (C2xC4):2D8  (C2xC6):8D8  (C2xC10):8D8  (C2xC14):8D8 ...

Matrix representation of C22:D8 ►in GL4(F17) generated by

1000
0100
0010
00016
,
1000
0100
00160
00016
,
14300
141400
00016
00160
,
141400
14300
0001
0010
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,0,16,0,0,16,0],[14,14,0,0,14,3,0,0,0,0,0,1,0,0,1,0] >;
 

C22:D8 in GAP, Magma, Sage, TeX 

C_2^2\rtimes D_8
 
% in TeX
 
G:=Group("C2^2:D8");
 
// GroupNames label
 
G:=SmallGroup(64,128);
 
// by ID
 
G=gap.SmallGroup(64,128);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,963,489,117]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C22:D8 in TeX

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